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Ultracold Quantum Gases Part 1: Bose-condensed Gases The experimentalist’s perspective Ultracold Quantum Gases Part 1: Bose-condensed Gases The experimentalist’s perspective Lecture 2

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Overview of course Lecture 1: Ultracold quantum gases: What? Why? How? Labtour Lecture 2: Atom-laser interaction Bloch sphere Lecture 3: Dressed state picture Optical Bloch equations Detecting an ultracold gas Lecture 4: Light forces Molasses cooling Sisyphus cooling Lecture 5: Atomic beam oven Zeeman slower Magneto-optical trap Technology for laser cooling Lecture 6: Optical dipole trap Magnetic trap Trap technology Evaporative cooling Characterizing a BEC Lecture 7: Characterizing a BEC Our research: - Laser cooling to BEC - Towards RbSr molecules Lecture notes and additional study material available on blackboard. Questions:

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Goals of the next three lectures Understand laser cooling and trapping Questions we want to answer: On our way to answering these questions, we‘ll also learn some general quantum mechanics (electron in EM field, density matrix) some quantum optics (Bloch sphere, optical Bloch equations) about Brownian motion how atomic clocks work (Ramsey spectroscopy) How does laser trapping work? How does laser cooling work? How cold can we get? Can we find improved schemes to get even colder? What are the optimal parameters for experiments? How dense can we get? What limits the timespan for which we can trap atoms?

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Meet the players The atom(s)The laser beam(s)The EM field modes

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The atoms Atoms used for laser cooling need to have Especially simple: atoms with noble gas shell + 1 valence electron: the alkalis a simple optical term scheme (avoid problem of optical pumping to dark state) optical transitions that are accessible by current laser technology

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Example: Rb atomic structure Lowest optical transition: [Kr].5s 1 [Kr].5p 1 s s s s s p p p d wavelength: Figure:

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Fine and hyperfine structure Angular momenta in 87 Rb: Fine structure valence electron orbital angular momentum nuclear spin [Kr].5s 1 [Kr].5p 1 valence electron spin e.g. electron magnetic moment in magnetic field from orbital motion Hyperfine structure e.g. nuclear magnetic moment in magnetic field of electron

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Two-level system For the moment, we approximate the internal structure of the atom by a two-level system: Example: 87 Rb (Rubidium) transition energy: energy spread of excited level:

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Meet the players The atom(s)The laser beam(s)The EM field modes

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The laser beam We approximate the laser beam by a plane wave with wavelength. unit vector pointing in direction of propagation electric field: magnetic field: magnitude wavevector: frequency: angular frequency: energy of photon: momentum of photon: intensity: wavelength wavevector electric field magnetic field direction of propagation typical intensity for laser cooling: Power of 1mW corresponds to ~10 15 photons per second (in the visible range of the spectrum).

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The laser beam A plane wave fills all of space. E-field at each location: Picture from Wikipedia.

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Phase front diagram Phase fronts:

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Meet the players The atom(s)The laser beam(s)The EM field modes

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In a finite volume (e.g. a hollow metal cube) only discrete modes of the EM field possible. Hollow metal cube with volume Two polarizations: e.g. linear with along or along Example shown: red lines symbolize sine term along respective direction n x = n y = 1 and n z =8 E-field zero at wall Eigenmodes: standing waves This field (with corresponding B-field) solves EM wave equation with where Each mode can be populated with photons. modes per unit frequency and unit volume: Mode density independent of chosen volume: The energy of each mode is (just like harmonic oscillator) quantum fluctuations The electromagnetic vacuum consists of these modes in their groundstate, for every mode.

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Basic atom-light interaction processes Absorption Stimulated emission Spontaneous emission

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Absorption photon incident on ground-state atom atom absorbes energy and momentum Example: stationary atom absorbing resonant photon properties of atom after absorption event: temperature corresponding to that energy, defined by It will be difficult to laser cool below the recoil temperature…

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Stimulated emission resonant laser beam populated by N photons incident on excited-state atom atom emits photon into laser mode N photonsN+1 photons momentum conservation lets atom recoil in direction oposite to photon momentum, i.e. oposite to laser beam direction

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Spontaneous emission Spontaneous emission excited-state atomatom spontaneously emits a photon into EM vacuum mode Probability of being in excited state Emission into opposite directions has equal probability. Momentum conservation lets atom recoil in direction oposite to photon momentum. Time ( ) Exponential decay on timescale :

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Fluorescence photons in laser beam laser beam photons & photons from stimulated emission scattered photons (spontaneous emission) Scattering rate Detuning Typical resonance in scattering rate Resonance often broadened by Doppler effect collisions saturation of resonance

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Fluorescence of strontium atomic beam Atomic beam oven Cooling laser Fluorescence

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Magneto-optical trap

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Our situation laser beams atom trajectory of atom other EM field modes Describe atom as particle with position and velocity. Atom experiences momentum kicks by absorption and emission of photons. Averaged over time, these momentum kicks correspond to a force. Calculate atomic trajectory using Newton‘s laws.

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Validity of approximation (I) In reality atom is a wavepacket with spatial spread and momentum spread. (neglecting factors of ) For particle approximation to be valid, the whole wavepacket has to be influenced in the same way. Light fields can change on a scale, therefore we require: For the whole momentum space wavepacket to be influenced in the same way, the linewidth of the transition needs to be larger than the Doppler shift. Condition (1) ; From Heisenberg: Condition (2) Condition of large linewidth Condition under which particle approximation is valid:

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Velocity change every Validity of approximation (II) For time averaged force to be sufficient to describe dynamics, the internal dynamics of the atom must be much faster than the external dynamics. Internal dynamics happens on timescale Relevant timescale for external dynamics: recoil kicks give rise to Doppler shift comparable to linewidth. Internal dynamics much faster than external dynamics: Condition of large linewidth Condition under which evolution is adiabatic:

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Condition of large linewidth Examples: The narrow laser-cooling transition of Sr has special laser cooling properties. narrow laser-cooling transition

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Calculation strategy laser beams atom trajectory of atom other EM field modes Step 1: consider position of atom fixed and describe internal evolution of atom Step 2: calculate force using the knowledge of the internal dynamics

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Step 1: internal state evolution For now: position of atom fixed. The atomThe laser beamThe field modes coupling Step 1a: only laser beam and atom: only absorption from laser beam and stimulated emission into that beam no spontaneous emission Bloch sphere picture & dressed state picture Step 1b: include EM field modes: also spontaneous emission optical Bloch equations Today‘s lecture Next lecture

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Nucleus and inner shell electrons create central potential on valence electron. Step 1a: laser beam & atom Assumptions and approximations Plane wave with linear polarization along propagating along approximated as classical field. E-field: Atom with a single valence electron, i.e. alkali atom. We can at the end of the calculation overcome most of the approximations concerning the atom by using experimentally determined atom-light interaction constants for a given transition. Laser beam: Atom: Neglecting spontaneous emission: Everything we will derive is valid for any two-level transition driven by sine perturbation, not only optical transitions driven by a monochromatic laser. Valid if time of experiment shorter than the lifetime of the excited level. Nucleus of atom fixed at origin. Only the single valence electron is influenced by the plane wave (hydrogen like behavior).

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Hamiltonian vector potential scalar potential electron charge electron mass electron spin operator Coulomb potential of nucleus and inner electrons on valence electron

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Dipole approximation Calculation steps and approximations Find gauge of EM field that simplifies Hamiltonian. Show that terms can be neglected. Show that interaction between electron magnetic moment and B field can be neglected. Note that spatial extend of electron wavefunction much smaller than wavelength of transition. Typical diameter of atom: < 1 nm Typical wavelength of atomic transition: several 100 nm Replace and by and. EM field that electron experiences can be approximated by the field at the location of the nucleus. Result see e.g. Sakurai 5.7, Cohen-Tannoudji A XIII,… (definitions vary by constant factors between books)

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Two-level approximation Eigenstates of are with energies. ; ; We want to find the eigenstates of and determine the dynamics of the coupled system. We assume that initially only and are populated and that the the energy between these states is much closer to the energy of the laser photons than any other transition. energy States besides and are barely populated during the dynamics and can be neglected. Proof: time-dependent perturbation theory with sine modulation or deduction from the following calculation

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Two-level system with sine perturbation energy General wavefunction: with normalization Goal: solve Schrödinger equation

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Matrix notation To obtain differential equations for and we need to determine matrix elements of Hamiltonian.

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Our case Our case: ; Matrix elements of Hamiltonian, e.g. Our Hamiltonian:

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Parity considerations 1D examples of functions with well-defined parity: even parity odd parity Integral over odd parity functions is zero. Parity of product of functions is product of parity of individual functions. (Not every function has well defined parity.) f 1 (x) f 2 (x)=x f 1 (x) f 2 (x) Parity is symmetry under reflection at the origin. even parity: odd parity:

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Our case Our case: ; Matrix elements of Hamiltonian, e.g. Our Hamiltonian: Parity considerations: Our bare atomic system is symmetric under parity since it consists of a particle in a spherically symmetric potential.

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Observation: and oscillate fast. Off-diagonal elements Our Hamiltonian: Schrödinger equation in matrix form: Define Rabi frequency: Schrödinger equation: (definition varies through literature) e.g. without laser:

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Basis change Basis change: Determine Schrödinger equation in new basis. Schrödinger equation in old basis: Put this rotation into state vector: (go into rotating frame)

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Set,. Use Our case Schrödinger equation: Schrödinger equation in rotated basis Schrödinger equation in old basis Rotating wave approximation: Neglect rapidly oscillating terms. Valid if Use

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Solve differential equations Solve this set of differential equations for atom in at laser phase with Probability of finding atom in :

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Resonant behavior On resonance: Time absorption dominates emission dominates Sine oscillations with period.

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Experiment: ion qubit

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Off resonant behavior Time on resonance = 0 off resonance off resonance Oscillations faster. Atom never fully in excited state.

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Resonance Time Excitation probability by square pulse with duration in dependence of detuning:

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The Bloch sphere Let‘s find an intuitive description. Observation: Express with two real parameters, and : with Can be represented as a point on a unit sphere!

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The Bloch vector picture We will show equation of motion with field vector Represent by Bloch vector Surface of Bloch sphere represents the Hilbert space of the two-level system. R. Feynman et al. J. Appl. Phys. 28, 49 (1957) © Wikipedia

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The Bloch vector picture Equation of motion of Bloch vector: The Bloch vector moves under the influence of a driving field in the direction orthogonal to the field vector and the Bloch vector itself. This is analog to the evolution of an angular momentum under the influence of a torque. © Wikipedia

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Schrödinger equation in rotated basis Schrödinger equation in old basis on with Result: Proof of Basis change using

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Proof of Show using Insert and into Show equality of left and right hand side using

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Both, and have a phase, and, respectively. Rotating frame Let‘s imagine the laser beam is not hitting the atom. The laser is still working, guaranteeing a well defined phase, but the beam is not sent on the atom. Solution: Compare to: Therefore: The Bloch vector is rotating around the north-south axis of the Bloch sphere if the atomic transition frequency is different from the laser frequency. The Bloch sphere is given in a frame rotating with the laser phase. What does that mean?

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Examples Switch laser on for as long as necessary to bring atom to excited state. : (different starting condition) For weaker laser, longer time needed.

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Rabi oscillations Compare to our calculation from before: Time

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Stationary states If Bloch vector is parallel or antiparallel to the Rabi vector, the atomic state only acquires a phase. These are the eigenstates of the system.

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Applications The Bloch vector picture gives intuitive insight into many applications. Nuclear magnetic resonance Quantum computation Atomic clocks Atom interferometers in general Everything we derived is valid for any two-level transition driven by EM wave. In the radiofrequency domain this is always the case on experimental timescales. In the optical domain this requires short laser pulses. Especially useful if pulse shorter than the lifetime of the excited level, so that spontaneous emission is negligible. Bloch vector picture can be generalized to include spontaneous emission and to describe an ensemble of many atoms.

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Atomic clock Definition of second: periods of the radiation corresponding to the transition between the cesium hyperfine levels Challenge: measure this transition precisely radiofrequency source atomic beam detector for feedback control Use Ramsey spectroscopy:

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Ramsey spectroscopy on resonance wait time result

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Ramsey spectroscopy off resonance wait time result Frequency difference between rf source and atomic transition leads to phase shift between and, in this example by.

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Ramsey spectroscopy signal LNE-SYRTE transition probability To determine if rf source is too low or too high in frequency, execute two measurements, with frequencies slightly above and below the rf source frequency.

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Fountain clock Atomic fountain clock, BNM-SYRTE, Paris NIST Long interrogation time (~1s) through slow (few cm/s), ultracold (few µK) atoms.

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Exercizes

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Exercize: -pulse condition 1) Determine with constant under the following starting conditions: Help: 2) How does the oscillation frequency depend on ? 3) How does the oscillation frequency depend on the intensity of the EM field? How much do you have to increase the laser intensity to double the oscillation frequency? 4) Assume that we start in and that the amplitude of is changing from zero to a finite value, back to zero. Give a condition on that needs to be fulfilled to execute a -pulse.

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Solution: -pulse condition ; 2) It doesn‘t. Crucial for spin echo (see later). 1) 3) The laser intensity has to be increased by a factor four to double the Rabi frequency. 4) The pulse area needs to be.

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Exercize: nuclear magnetic resonance Was explained during exercize part of class See the web (e.g. Wikipedia or ) for an explanation of NMR. Here the question (in short. Longer version given in class): You have an ensemble of two level systems, each with a slightly different transition frequency. After a -pulse, the spins will fan out on the equatorial plane of the Bloch sphere. How can you resynchronize their Bloch vectors?

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Solution: spin echo Picture from Wikipedia.

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